Buddhabrot

Robert P. Munafo, 2008 Feb 12.

Name given by Lori Gardi to the plotting technique invented by Melinda
Green, which renders the Mandelbrot Set in a way similar to the
Inverse-Iteration Method but using forward iteration and plotting
the iterates in parameter space. If viewed sideways (with West
at the top and North to the right) the Continent appears a bit
like certain images of Buddha.

To produce a Buddhabrot image, pick a dwell limit as usual, and
select a view that includes the entire Mandelbrot Set. Then, iterate
as you would for a Mandelbrot image (see iteration algorithm), and
find out if the iteration escapes. If it escapes, then look at every
value of Z that occurred to see which pixel that Z value falls in
(some Z values will be outside the view). Keep a count for each pixel,
recording how many times a Z value fell within that pixel. After all
the points have been iterated (or occasionally while iterating) plot
all the pixels, choosing a color for each pixel based on the recorded
count.

If you want to view a low-resolution image quickly, you have to choose
points (values of C) to iterate in some way other than a normal
grid scan. Melinda Green, Paul Bourke and Marc Jaouen choose points
at random, however this produces a noisy "snow" effect that makes it
necessary to do much more iteration before the subtle details of the
Buddhabrot image become visible. A grid scan, or an interleaved
stagged grid scan like that used in ordered dithering will reduce
(but not eliminate) the noise.

The basic Buddhabrot image is similar to a view of the entire
4-dimensional Julia-Mandelbrot Space collapsed to a single plane
(the Z plane), and showing all the Iterates of Z0=0, but no
iterates for any other initial Z. This shows the orbits, which
contain different numbers of attracting points depending on the
periods of the individual orbits. However because the bounded
orbits do not get plotted, the Buddhabrot image only contains iterates
of 0 for the Fatou dusts. The points that have high dwell values
dominate, and their orbits include many points that fall near the
value of C used for that iteration. The C values of points with high
dwells are the points near the boundary of the Mandelbrot set, and
that is why the Buddhabrot image contains an image of the Mandelbrot
set.

There are also many rotated and enlarged copies in different places.
All of the rotated and enlarged copies are preimages under the
iteration of the escaping C values. The number of copies depends on
the various periods of the orbital dynamics near the points being
iterated. For example, there is one extra image of the area around
R2.1/2a, two extra images each of the areas around R2.1/3a and
R2.2/3a, and so on. The images of the smaller mu-atoms are fainter
because fewer iterates contribute to them. Because of the Squaring
that occurs to the iterate once per period in its iteration, many of
the preimages have the characteristic shape of a Mu-molecule, a
cardioid with mu-unit decorations, and the cardioid's cusp is
near the origin.

See also: Web pages by Melinda Green, Lori Gardi, Marc Jaouen and/or
Paul Bourke. (To find any of these, use a search engine to search for
these names or for "Buddhabrot")